552 The two-thirds rule of Bézier curves

- Hey gang! This is Deke McClelland. Welcome to Deke's Techniques. Now over the last couple of weeks we've been talking about Bezier curves inside the various Adobe programs whether it's Illustrator, Photoshop, Indesign, what have you. But no amount of theory is gonna teach you how to draw. And one of the frustrations that I hear from folks over and over again is that they feel like they can't reliably create smooth curves inside these programs. Now I'll be covering a few rules in my upcoming course Adobe Pen Tool Mastery.

But today I'm gonna review one of the oldest rules there is that goes by the name The Two-Thirds Rule. And the idea is that both of the control handles together take up about two-thirds of the entire length of a curving segment. Here, let me show you in extraordinary detail exactly how it works. Alright, let's take a look at The Two-Thirds Rule as it applies to Bezier curves whether you're working inside Illustrator, Indesign, Photoshop or any other Adobe application. Now most folks know this as The Two-Thirds Rule.

But it's also known as The Rule of Thirds or what I call The One-Third Gap Rule for reasons that will become obvious later inside this movie. And so, let's start by considering the perfect curve which is a circle. Or more specifically a single segment in this circle which is to say a quarter circle or an arc. And as you may know an arc begins and ends at an anchor point. And we also have two control handles that are pulling at it in order to make it curve.

So, just rough eye-balling things this lever, that is the imaginary line that connects the circular control handle to the square anchor point, is about one-third. And this lever down here is about one-third as well. One-third of what you might ask? And the answer is the total length of the segment. Meanwhile the distance between the two control handles, which is known as the invisible lever, as you may recall from our technical look at Bezier curves a couple of weeks ago, consumes the remaining third.

And so, if I were to share with you the approximate dimensions then the length of the curve is about 800 points, and this is true, by the way, inside of this particular diagram, while the length of each one of the visible levers is about 280 points and the length of the invisible lever is 320 points. So, notice I've gone ahead and super imposed one of the visible levers on to the invisible one so that you can see that the visible one is slightly shorter. And so, if you were to do the math here, if you were to add 280 points to 320 to 280 you would get a total of 880 points which is more than the length of the segments.

So, this is just an approximation of how things work. But when everything is said and done each one of these levers including the invisible one consumes about a third of the entire length of the segment which is why this is sometimes known as The Rule of Thirds. And while that's absolutely beautiful that things work this way it's a little bit misleading as well. After all, how do you explain this curve right here which is an absolutely smooth curve. It's perfectly fine. And yet the top lever is obviously much longer than the bottom one.

And this becomes even more obvious where this curve is concerned. This guy up top is more than twice as long as the one below. And so what we've got here, just to give you a sense, is a lever up top that's about half the total length of the segment. And this other down here that's just one-sixth the length. And so, the math becomes a little more complicated this time. You may know that one-half is the same as three-sixths So, if we take three-sixths and add it to one-sixth we get four sixths which we can factor down to two-thirds, hence, The Two-Thirds Rule.

So, as long as the two segments add up to two-thirds the total length of the segment that's what actually counts. But really seriously who on Earth is gonna do this kind of math especially in their heads while they're drawing Bezier curves, for example, with the pen tool? Well, that's why I urge you to forget all about that stuff. And pay attention instead to the invisible lever. And so, I want you to watch what's going on here. Keep an eye on the invisible lever throughout each one of these diagrams.

Here we've got a perfectly acceptable smooth curve. The same is true for this curve right here which is the quarter circle, the perfect arc. We've also got this guy which is a little bit askew. Then we've got this one with the very long top lever. But we could go with the very long left-hand lever instead. Or we could switch things to the other side. And we could even move one control handle on the opposite side of the segment from the other one. But in each case are you noticing what's going on with the invisible lever that is the dotted red line? In each case it is remaining the same length.

And that length is one-third of the segment. Which is why I call this The One-Third Gap Rule. Because it doesn't really matter that much how long the visible levers are. What matters most is the length of the invisible lever between the two round handles that are controlling the curvature of the segment. Now while I'm big fan of this rule it really does work. You wanna resist taking it too far. Notice in this case with this very long lever up top and this very short one down below that we are ending with a weird curl at the end of this segment.

And the truth is if we take this rule, The One-Third Gap Rule to it's logical extreme, then we're gonna end up with a single lever that consumes two-thirds the length of the entire segment. So, in this case we've got one two-thirds visible lever. We still have the invisible lever taking up one-third. But we're missing any control handle at the point where the segment enters this anchor point. And as a result, we have this kind of flat tire of a segment. And ironically we've got curvature where we have no control handle.

But we've got flatness where we do have a control handle. And if you have any doubt for the need of two control handles per segment, because I know a lot of people do. There's a lot of people out there that'll just sort of sketch one handle at a time which is typically a bad idea. And the reason is you get flattened out curves like this one right here. And just so we can see what's going on in the background here, these are the anchor points right here and these are the single control handles associated with each and every segment which frankly, is an absolute no-no.

You can either have no control handle if you want a straight segment. But if you wanna curved segment, you need two control handles per segment. And so, I just want you to see one of the problems with this ellipse is not only this flatness up here at the top and then over here on the side and so forth, but we also have slight corners at each one of the anchor points as opposed to smooth transitions. Compare that with an ellipse that is drawn with two control handles per segment that follows the One-Third Gap Rule, as you can see right here, and we end up with a very smooth ellipse indeed.

And this One-Third Gap Rule follows, by the way, even if we move around the anchor points like crazy. So, in this case we no longer have a bunch of levers that are all the same length, instead we have a bunch of differently scaled levers as well as a bunch of differently scaled segments. So, notice the segment up top here is much shorter than this one down below. But as a result the top levers are shorter and the bottom levers are longer. And the result is that we get an extremely smooth curve all the way around.

And this goes, by the way, whether you're working with smooth points as we've been seeing so far or cuss points. So, in this case we've got a bunch of corners with independent control handles going in different directions but we're still subscribing to the One-Third Gap Rule. And the result is that we get these smoothly transitioning segments that meet at definite corners as we're seeing here without all the interface falderal. And so what really counts is that we have smooth, fluid, organic curves.

And so, my ultimate advice here is that you keep the gap at about one-third the length of the segment as we've been seeing so far. And that you don't let either lever extend past one-half the length of that segment. And so, this is about as extreme as you want things to be. And that, my friends, is The Two-Thirds Rule, also known as The Rule of Thirds, or what I like to call The One-Third Gap Rule as it applies to drawing and editing Bezier curves inside of Illustrator, Indesign, Photoshop and other Adobe applications.

Alright, so much for this week. Next week I'm gonna show you how to create a neon sign entirely from scratch inside Adobe Illustrator. Deke's Techniques each and every week. Keep watching.